Symmetry and monotonicity of least energy solutions

Abstract

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153-179, 1988) and MariAY (Arch Ration Mech Anal, 192:311-330, 2009) and answer questions from Br,zis and Lieb (Comm Math Phys 96:97-113, 1984) and Lions (Ann Inst H Poincar, Anal Non Lin,aire 1:223-283, 1984).